> heat isn't real, it's just a measurement of the entropy in a system

You say that as if entropy ain't realy.

> if hot water takes shortcuts and freezes faster than cold, then it requires less energy to do so - that is the crux of the debate

You can't escape the first and second laws of thermodynamics.

It took energy both ways though, first to heat the water and then freeze it. Pretty sure the final bill will not show less energy used.